Is the wife obligated to sell her personal gold to pay off her husband's loan that he took for marriage, especially since he has repented and regrets his action, and would she be sinful if she did not do so?

You are not obligated to sell your gold to pay off your husband's debt, and there is no sin upon you for keeping it. You have fulfilled your duty by urging him to repent from borrowing with interest. Repentance erases what came before it, and one who repents from a sin is like one who never sinned.